Ehrenfest equations

Ehrenfest equations (named after Paul Ehrenfest) are equations which describe changes in specific heat capacity and derivatives of specific volume in second-order phase transitions. The Clausius–Clapeyron relation does not make sense for second-order phase transitions, as both specific entropy and specific volume do not change in second-order phase transitions.

Ehrenfest equations are the consequence of continuity of specific entropy ${\displaystyle s}$ and specific volume ${\displaystyle v}$, which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy ${\displaystyle s}$ as a function of temperature and pressure, then its differential is: ${\displaystyle ds=\left({{\partial s} \over {\partial T}}\right)_{P}dT+\left({{\partial s} \over {\partial P}}\right)_{T}dP}$. As ${\displaystyle \left({{\partial s} \over {\partial T}}\right)_{P}={{c_{P}} \over T},\left({{\partial s} \over {\partial P}}\right)_{T}=-\left({{\partial v} \over {\partial T}}\right)_{P}}$, then the differential of specific entropy also is:

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